“Uniqueness in the small” entitles Section 4.2 of the monograph [1]} co-authored by Olga A. Ladyzhenskaya and Nina U. Uralceva.
This work is especially renowned for providing a complete uniqueness analysis of quasilinear pde's in the scalar case.
In the present setting, the following approach does not seem too presumptuous:
we could transfer the uniqueness in the small result from p.257 of [1]} to the case of our system just following the same ideas and adding more and more restrictions when it becomes necessary.
Bear in mind that a (nonlocal) uniqueness result is provided in [3]} in the case \(\Omega \subset \mathbb {R}^2\)  as soon as a first restriction on the ratios between cross-diffusive and diffusive parameters ensures the additional regularity in \(L^4(0,T;W^{1,4}(\Omega ))\)  of the solution and as another technical restriction is assumed.
We now aim at checking if a local uniqueness result could be reached with weaker assumptions.
The computations are detailed in the following lines.
Notice that they also shed light on the assumptions that could lead to a result of overall uniqueness when \(N>2\) .
